The necessity of the sausage-string structure for mode-locking regions of piecewise-linear maps
| dc.citation.volume | 462 | |
| dc.contributor.author | Simpson DJW | |
| dc.date.accessioned | 2024-10-23T02:12:33Z | |
| dc.date.available | 2024-10-23T02:12:33Z | |
| dc.date.issued | 2024-06 | |
| dc.description.abstract | Piecewise-smooth maps are used as discrete-time models of dynamical systems whose evolution is governed by different equations under different conditions (e.g. switched control systems). By assigning a symbol to each region of phase space where the map is smooth, any period-p solution of the map can be associated to an itinerary of p symbols. As parameters of the map are varied, changes to this itinerary occur at border-collision bifurcations (BCBs) where one point of the periodic solution collides with a region boundary. It is well known that BCBs conform broadly to two cases: persistence, where the symbolic itinerary of a periodic solution changes by one symbol, and a nonsmooth-fold, where two solutions differing by one symbol collide and annihilate. This paper derives new properties of periodic solutions of piecewise-linear continuous maps on Rn to show that under mild conditions BCBs of mode-locked solutions on invariant circles must be nonsmooth-folds. This explains why Arnold tongues of piecewise-linear maps exhibit a sausage-string structure whereby changes to symbolic itineraries occur at codimension-two pinch points instead of codimension-one persistence-type BCBs. But the main result is based on the combinatorical properties of the itineraries, so the impossibility of persistence-type BCBs also holds when the periodic solution is unstable or there is no invariant circle. | |
| dc.description.confidential | false | |
| dc.edition.edition | June 2024 | |
| dc.identifier.citation | Simpson DJW. (2024). The necessity of the sausage-string structure for mode-locking regions of piecewise-linear maps. Physica D: Nonlinear Phenomena. 462. | |
| dc.identifier.doi | 10.1016/j.physd.2024.134142 | |
| dc.identifier.eissn | 1872-8022 | |
| dc.identifier.elements-type | journal-article | |
| dc.identifier.issn | 0167-2789 | |
| dc.identifier.number | 134142 | |
| dc.identifier.pii | S0167278924000939 | |
| dc.identifier.uri | https://mro.massey.ac.nz/handle/10179/71829 | |
| dc.language | English | |
| dc.publisher | Elsevier BV | |
| dc.publisher.uri | https://www.sciencedirect.com/science/article/pii/S0167278924000939 | |
| dc.relation.isPartOf | Physica D: Nonlinear Phenomena | |
| dc.rights | (c) 2024 The Author/s | |
| dc.rights | CC BY 4.0 | |
| dc.rights.uri | https://creativecommons.org/licenses/by/4.0/ | |
| dc.subject | Piecewise-smooth | |
| dc.subject | Border-collision bifurcation | |
| dc.subject | Invariant circle | |
| dc.subject | Arnold tongue | |
| dc.title | The necessity of the sausage-string structure for mode-locking regions of piecewise-linear maps | |
| dc.type | Journal article | |
| pubs.elements-id | 488049 | |
| pubs.organisational-group | Other |